For any two nonzero p-adic numbers a and b, the Hilbert symbol is defined as (a, b) = {1 | if z^2 = a x^2 + b y^2 has a nonzero solution - 1 | otherwise. auto right match If the p-adic field is not clear, it is said to be the Hilbert symbol of a and b relative to k. The field can also be the reals (p = ∞). The Hilbert symbol satisfies the following formulas: 1.(a, b) = (b, a). 2.(a, c^2) = 1 for any c. 3.(a, - a) = 1.

Diophantine equation--2nd powers | field | p-adic number | symmetric bilinear form | vector space